Everyone knows at least five companies you should contact. Even Aunt Mabel.
Ask your network contacts the following question: "Which are the top five companies that you recommend I contact?" Many people are able to give a "Top Five" (or at least a "Top Three") list quite easily. After they give you the names, ask them if they have any personal contacts at any of those companies. Keep in mind that over time you will begin to hear some of the same company names being repeated (especially if they are well-known industry leaders). But keep asking for the names of new personal contacts—you can never have enough. After they give you the contact names, selectively consider (depending on your comfort level with the contact) asking if they will contact the company on your behalf. Sometimes your network contact can do some of the work for you! When they make the referrals, be certain that you follow through in a professional manner since they are putting their personal/professional reputation on the line for you. Attaching their name with your name puts a heavy responsibility on you to meet or exceed all expectations.